Problem 37 - Calculus and its applications

Question

Solve by separating variables. \(3 y^{2} \frac{d y}{d x}=8 x\)

Step-by-Step Solution

Answer

\( y = -\frac{3}{4 x^2 + 3C} \) is the general solution.

Step 1: Separate the Variables

To solve the differential equation \( 3 y^{2} \frac{d y}{d x} = 8 x \) by separating variables, begin by dividing both sides by \( 3y^{2} \). This separates the variables: \( \frac{d y}{y^2} = \frac{8}{3} x \, d x \).

Step 2: Integrate Both Sides

Integrate both sides of the separated equation. On the left, integrate \( \int \frac{1}{y^2} \, d y \), and on the right, integrate \( \int \frac{8}{3} x \, d x \). The integrals are: \( -\frac{1}{y} = \frac{4}{3} x^2 + C \), where \( C \) is the constant of integration.

Step 3: Solve for \( y \)

Solve for \( y \) by first taking the reciprocal of both sides and then simplifying. Multiply both sides by -1 to get \( \frac{1}{y} = -\frac{4}{3} x^2 - C \) and then take reciprocals: \( y = -\frac{1}{\frac{4}{3} x^2 + C} \).

Step 4: Final Simplification

Simplify the expression for \( y \) if possible. Since this is a general solution, express it as \( y = -\frac{3}{4 x^2 + 3C} \) by letting \( C' = 3C \).

Key Concepts

Differential Equations

Differential equations are fundamental in understanding how variables change with respect to one another. In many cases, these equations model real-world phenomena, such as how populations grow or how heat diffuses through a material. A differential equation involves a function and its derivatives, expressing a relationship between them.

There are various types of differential equations. The simplest kind is the ordinary differential equation (ODE), which we see in this exercise. They involve functions of a single variable and their derivatives. In our case, the equation is structured to describe how the variable \(y\) changes in response to \(x\).

• Order: This refers to the highest derivative present in the equation. Here, as \(\frac{d y}{d x}\) is the highest derivative, it is a first-order differential equation.
• Linearity: Linear equations are those where both the function and its derivative appear to the first power. However, when powers of a function (like \(y^2\) in this problem) are involved, it's non-linear.

Many real-world problems cannot be expressed in simple algebraic equations, making differential equations invaluable tools in science and engineering.

Integration Techniques

Integration is a central operation in calculus, essential for solving differential equations. When solving a differential equation by separation of variables, as in this exercise, integration helps us find an expression for the function describing the relationship between variables.

Here, after you separate the variables and ensure \(dy\) terms are on one side and \(dx\) terms are on the other:

• The left side becomes \(\int \frac{1}{y^2} \, dy\),
• The right side becomes \(\int \frac{8}{3} x \, dx\).

Both integrals are computed as follows:

• For \(\int \frac{1}{y^2} \, dy\): Recognize it as \(\int y^{-2} \, dy\), which results in \(-\frac{1}{y}\).
• For \(\int \frac{8}{3} x \, dx\): This is a straightforward polynomial integral resulting in \(\frac{4}{3} x^2\).

Don't forget to add the constant of integration (\(C\)). Constants are crucial as they account for the family of solutions contained in indefinite integrals.

Calculus Problem Solving

Solving calculus problems, particularly those involving differential equations, requires a systematic approach. Let's break down the problem-solving strategy used here:

1. **Recognize the Type of Equation**: Understand that separating variables is suitable for the structure of our differential equation.
2. **Separate the Variables**: Move terms involving \(y\) to one side and terms involving \(x\) to the other, ensuring you've organized the equation properly for integration.
3. **Integrate**: Apply integration techniques to both sides separately, ensuring you integrate with respect to the right variable.
4. **Solve for the Function**: Once integrated, you have a function of \(y\) in terms of \(x\). Solve for \(y\) explicitly, if possible, to find the general solution.

This process not only answers the question but builds a conceptual framework useful for tackling various differential equations. Practice and intuition are your best allies in mastering these techniques. Always question your work for consistency and logical progress, especially in the choice of separation and integration methods.